As a teacher I found myself often asking the question, "Are you even in the ballpark?" meaning is your answer close. Many of my students didn't know if I mean Wrigley Field or Yankee Stadium. If a student has no number sense and can not visually see what the answer should be about, there is no way they will be able to know if there answer is even close. Students spend a large majority of their math classes memorizing vocabulary, rules, and procedures with little understanding. When the brain is focusing on memorizing procedures it often doesn't take into account the size of the number or the common sense approach to solve the problem. Let's look at an example:
When I taught second grade my students had no problems solving the problem 1,000 - 997. They would simply state, "It's just three away: 998, 999, 1,000". When a fourth grade teachers walked into my room and saw this discussion she was shocked. She claimed her fourth grade students couldn't even solve this problem. I don't think my second graders were smarter than the fourth graders, I think something very important happened between 2nd and 4th grades.
My second graders were looking at this problem conceptually - the distance between the two numbers. The fourth graders were too focused on trying to remember a procedure. They lined up the numbers vertically and were so focused on remembering, when borrowing does that 0 become a 10 or a 9 or a 8 as I borrow across the columns. When memorizing procedure, students often forget to use their common sense. Subtraction can be thought of as the distance between two numbers.
As teachers, it is important to balance conceptual understanding and learning of procedures. Research states we need to build conceptual understanding first, and through this students will understand better the procedural. NCTM - National Council of Teachers of Mathematics believes that if we have true number sense we will develop our own procedures for adding, subtracting, multiplying, and dividing that is both efficent and accurate. I have sen this over and over with students who have used a research based strategy called Cognitvely Guided Instruction. Through CGI students build number sense and often create algorithms (procedures) that make sense to them. They continue to "think" about numbers and how numbers work together. I thank all teachers who focus their students into the "thinking" about mathematics. Thank you!
What are your thoughts?
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